Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $y = \dfrac{7(2p - 9)}{4} \div \dfrac{12p^2 - 54p}{9} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{7(2p - 9)}{4} \times \dfrac{9}{12p^2 - 54p} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 7(2p - 9) \times 9 } { 4 \times (12p^2 - 54p) } $ $ y = \dfrac {9 \times 7(2p - 9)} {4 \times 6p(2p - 9)} $ $ y = \dfrac{63(2p - 9)}{24p(2p - 9)} $ We can cancel the $2p - 9$ so long as $2p - 9 \neq 0$ Therefore $p \neq \dfrac{9}{2}$ $y = \dfrac{63 \cancel{(2p - 9})}{24p \cancel{(2p - 9)}} = \dfrac{63}{24p} = \dfrac{21}{8p} $